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ω-stable
In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify. Stable theories were the predominant subject of pure model theory from the 1970s through the 1990s, so their study shaped modern model theory and there is a rich framework and set of tools to analyze them. A major direction in model theory is "neostability theory," which tries to generalize the concepts of stability theory to broader contexts, such as simple and NIP theories. Motivation and history A common goa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stability Spectrum
In model theory, a branch of mathematical logic, a complete theory, complete first-order theory ''T'' is called stable in λ (an infinite cardinal number), if the Type (model theory)#Stone spaces, Stone space of every structure (mathematical logic), model of ''T'' of size ≤ λ has itself size ≤ λ. ''T'' is called a stable theory if there is no upper bound for the cardinals κ such that ''T'' is stable in κ. The stability spectrum of ''T'' is the class of all cardinals κ such that ''T'' is stable in κ. For countable theories there are only four possible stability spectra. The corresponding dividing line (model theory), dividing lines are those for totally transcendental theory, total transcendentality, superstable theory, superstability and stable theory, stability. This result is due to Saharon Shelah, who also defined stability and superstability. The stability spectrum theorem for countable theories Theorem. Every countable complete first-order theory ''T'' falls into o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forking Extension
In model theory, a forking extension of a type is an extension of that type that is not whereas a non-forking extension is an extension that is as free as possible. This can be used to extend the notions of linear or algebraic independence to stable theories. These concepts were introduced by S. Shelah. Definitions Suppose that ''A'' and ''B'' are models of some complete ω-stable theory ''T''. If ''p'' is a type of ''A'' and ''q'' is a type of ''B'' containing ''p'', then ''q'' is called a forking extension of ''p'' if its Morley rank In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model theory, model of a theory (logic), theory, generalizing the notion of dimension in algebraic geometry. Definition Fix a theory ''T'' with a ... is smaller, and a nonforking extension if it has the same Morley rank. Axioms Let ''T'' be a stable complete theory. The non-forking relation ≤ for types over ''T'' is the unique relation that sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absoluteness (logic)
In mathematical logic, a formula is said to be absolute to some class of structures (also called models), if it has the same truth value in each of the members of that class. One can also speak of absoluteness of a formula ''between'' two structures, if it is absolute to some class which contains both of them. Theorems about absoluteness typically establish relationships between the absoluteness of formulas and their syntactic form. There are two weaker forms of partial absoluteness. If the truth of a formula in each substructure ''N'' of a structure ''M'' follows from its truth in ''M'', the formula is downward absolute. If the truth of a formula in a structure ''N'' implies its truth in each structure ''M'' extending ''N'', the formula is upward absolute. Issues of absoluteness are particularly important in set theory and model theory, fields where multiple structures are considered simultaneously. In model theory, several basic results and definitions are motivated by absolutene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrzej Ehrenfeucht
Andrzej Ehrenfeucht (, born 8 August 1932) is a Polish-American mathematician and computer scientist. Life Andrzej Ehrenfeucht formulated the Ehrenfeucht–Fraïssé game, using the back-and-forth method given in Roland Fraïssé's PhD thesis. Also named for Ehrenfeucht is the Ehrenfeucht–Mycielski sequence. In 1971 Ehrenfeucht was a founding member of the Department of Computer Science at the University of Colorado at Boulder. He currently teaches and does research at the University, where he runs a project, "breaking away", with Patricia Baggett; the project, using hands-on activities, aims at raising high-school students' interest in mathematics and technology. Two of Ehrenfeucht's students, Eugene Myers and David Haussler, contributed to the sequencing of the human genome. They, with Harold Gabow, Ross McConnell, and Grzegorz Rozenberg, spoke at a 2012 University of Colorado two-day symposium honoring Ehrenfeucht's 80th birthday. Two journal issues have come out in h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half Graph
In graph theory, a branch of mathematics, a half graph is a special type of bipartite graph. These graphs are called the half graphs because they have approximately half of the edges of a complete bipartite graph on the same vertices. The name was given to these graphs by Paul Erdős and András Hajnal. Definition To define the half graph on 2n vertices u_1,\dots u_n and v_1,\dots v_n, connect u_i to v_j by an edge whenever i\le j. The same concept can also be defined in the same way for infinite graphs over two copies of any ordered set of vertices. The half graph over the natural numbers (with their usual ordering) has the property that each vertex v_j has finite degree, at most j. The vertices on the other side of the bipartition have infinite degree. Properties Distances In a half graph, every two vertices are at distance one, two, or three. Any two vertices u_i and u_j are at distance two via a path through v_n, and any two vertices v_i and v_j are at distance two via a pat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. Other properties related to completeness The property converse to completeness is called soundness: a system is sound with respect to a property (mostly semantical validity) if each of its theorems has that property. Forms of completeness Expressive completeness A formal language is ''expressively complete'' if it can express the subject matter for which it is intended. Functional completeness A set of logical connectives associated with a formal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Independence
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non- trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically independent over K if and only if \alpha is transcendental over K. In general, all the elements of an algebraically independent set S over K are by necessity transcendental over K, and over all of the field extensions over K generated by the remaining elements of S. Example The real numbers \sqrt and 2\pi+1 are transcendental numbers: they are not the roots of any nontrivial polynomial whose coefficients are rational numbers. Thus, the sets \ and \ are both algebraically independent over the rational numbers. However, the set \ is ''not'' algebraically independent over the rational numbers \mathbb, because the nontrivial polynomial :P(x,y)=2x^2-y+1 is zero when x=\sqrt and y=2\pi+1. Algebraic independence of known constants Although an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Independence
In the theory of vector spaces, a set (mathematics), set of vector (mathematics), vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of Dimension (vector space), dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space. Definition A sequence of vectors \mathbf_1, \mathbf_2, \dots, \mathbf_k from a vector space is said to be ''linearly dependent'', if there exist Scalar (mathematics), scalars a_1, a_2, \dots, a_k, not all zero, such that :a_1\mathbf_1 + a_2\mathbf_2 + \cdots + a_k\mathbf_k = \mathbf, where \mathbf denotes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
